1. CURVES

2. CURVES
WHY??
WHAT??
PARAMETERS??

3. NECESSITY ARISES DUE TO PHYSICAL & GEOGRAPHICAL FEATURES
NECESSARY EVIL
CURVE IS A LINE OF WHICH NO PART IS STRAIGHT AND WHICH CHANGES DIRECTION WITHOUT ANGLES
POSITIVE IMPEDIMENTS FOR HIGHER SPEEDS

4. ANGULARITY OF AXLE WHILE NEGOTIATING A CURVE
Angle of attack α
PLAY HELPS THE WHEEL NEGOTIATE CURVE
A BOGIE WHICH CAN TAKE RADIAL POSITION ON THE CURVES IS BETTER
GUIDANCE SHALL BE FROM THE TRACK GUIDANCE AND NOT FLANGE CONTACT

5. CURVE DESIGNATION
CURVES ARE DESIGNATED BY THEIR RADII, EXCEPT ON IR & US RAIL ROADS
ON IR DEGREE OF CURVE FOR DESIGNATION
RADII FOR CALCULATION

6. DEGREE OF CURVE IS THE ANGLE SUBTENDED BY 30.5m CHORD AT THE CENTRE
100 feet=30.5m
D = 1750/R D R

7. V*(2R-V) = C/2*C/2 2RV=C2/4 V = C2/8R
ON IR, THE CURVES ARE MEASURED BY VERSINE- WHICH IS MID CHORD OFFSET ON 20m CHORD
V*(2R-V) = C/2*C/2
2RV=C2/4
V = C2/8R 2R V C

8. PARAMETERS OF CURVES RADIUS, R AND DEGREE OF CURVE, D ACTUAL CANT Ca
CANT DEFICIENCY Cd
CANT EXCESS Cex
EQUILIBRIUM CANT Ce / EQUILIBRIUM SPEED Ve
RATE OF CHANGE OF ACTUAL CANT rca
RATE OF CHANGE OF CANT DEFICIENCY rcd
CANT GRADIENT, i
LENGTH OF TRANSITION, L

10. EFFECTS OF CURVES
VEHICLE RUNNING AT A SPEED OF V IN A CURVE OF RADIUS R EXPERIENCES A CENTRIFUGAL FORCE = MV2/R
UNDESIRABLE EFFECTS
POSSIBLE PASSENGER DISCOMFORT
POSSIBLE DISPLACEMENT OF LOADS
RISK OF VEHICLE OVERTURNING

11. EFFECTS OF CURVES RISK OF DERAILMENT
HIGH LATERAL FORCE RESULTING IN MAINTENANCE PROBLEMS
CURVE RESISTANCE
WEAR OF RAIL & WHEEL FLANGE
LATERAL FORCE ON TRACK STRUCTURE

12. COMPENSATION FOR CURVATURE ON GRADIENT
COMPENSATION ALLOWED ON GRADIENTS DUE TO CURVATURE
70/R
0.04% PER DEGREE

13. SUPERELEVATION / CANT
THE EFFECT OF CENTRIFUGAL FORCE IS ELIMINATED/REDUCED BY RAISING THE OUTER RAIL BY A SPECIFIED AMOUNT.
THIS RAISING OF OUTER RAIL OVER INNER RAIL IS CALLED SUPERELEVATION/CANT
THE FORCE DUE TO THE RAISING OF THE OUTER RAIL IS EXERTED INWARDS AND IS CALLED CENTRIPETAL FORCE

14. VEHICLE ON A CANTED TRACKG SE
Centrifugal Force   W
Centripetal Force Wsinθ 

15. EQUILIBRIUM CANT
WHEN ON CIRCULAR MOTION , IF THE RESULTANT OF WEIGHT & CENTRIFUGAL FORCE IS PERPENDICULAR TO THE PLANE OF RAIL & PASSES THROUGH THE CENTRE OF TRACK, THE CORRESPONDING SPEED IS CALLED EQUILIBRIUM SPEED & THE CANT IS CALLED EQUILIBRIUM CANT

16. EQUILIBRIUM CANT IRPWM STIPULATION
EQUILIBRIUM SPEED IS TO BE DECIDED BY CE CONSIDERING
MAX. SPEEDS OF FAST & SLOW MOVING TRAINS
PERMANENT SPEED RESTRICTION
JUNCTIONS
STOPPING PLACES
GRADIENT AFFECTING SPEED OF GOODS TRAIN

17. Equilibrium Speed Russian Formula:
ni: no of trains, wi: weight of such train, Vi: speed of such train, m: types of trains

18. ACTUAL CANT Ca MAINTENANCE CRITERIA OVERTURNING AT INNER RAIL
HIGH CANT WILL CAUSE ROLLING OF BALLAST & FLATTENING OF INNER RAIL
OVERTURNING AT INNER RAIL
NOT VERY SENSITIVE TO WIND FORCE
SAFETY AGAINST DERAILMENT
EMPTY WAGON STOPPED & STARTED
COMFORT CRITERIA
NO APPRECIABLE DISCOMFORT UPTO 180 mm
LIMITED TO 1/8 TO 1/10 OF DYNAMIC GAUGE

19. MAXIMUM VALUE OF CANT

20. IRPWM PROVISIONS EQUILIBRIUM CANT (406(1)(a))
C=GV2/(127 R)
MAXIMUM CANT (406(1)(d)(i))
165 MM FOR GROUP A, B AND C ROUTE
185 MM FOR STRUCTURES
140 MM FOR GROUP D AND E ROUTE

21. Cant Deficiency

22. CANT DEFICIENCY Cd SAFETY COMFORT CRITERIA
UPTO 175 mm SAFE WITH CRITICAL WIND VELOCITY
COMFORT CRITERIA
DISCOMFORT IF UNBALANCED LATERAL ACCELERATION IS GREATER THAN 0.1g
CANT DEFICIENCY SHOULD BE LESS THAN 0.1G
OBSERVED VALUE OF ULA IS MORE THAN THE THEORETICAL VALUE

23. MAXIMUM VALUE OF CANT DEFICIENCY

24. Cant Excess

25. CANT EXCESS Cex NO COMFORT CONSIDERATION MAINTENANCE CONSIDERATION
EXCESS WEAR ON INNER RAIL
MORE THE VOLUME OF GOODS TRAFFIC LESSER WOULD BE CANT EXCESS
WORKED OUT FOR BOOKED SPEED OF GOODS TRAINS.
NORMALLY 65 KMPH ON BG.

26. Cant Excess values SNCF DB IMPORTANCE TONNAGE PER DAY CANT EXCESS
VERY IMP.
>45000 70
IMPORTANT
25,000 TO 45,000 80
MEDIUM
10,000 TO 25,000 90
UNIMPORTANT
<10,000
100
DB 
>60,000 50
30,000 TO 60,000
10,000 TO 30,000
110

27. IRPWM PROVISIONS CANT DEFICIENCY(406(2)) CANT EXCESS (406(3))
FOR GROUP A AND B ROUTES FOR SPEEDS IN EXCESS OF 100 KMPH FOR NOMINATED STOCK RUNS WITH PERMISSION OF CE : 100 MM
FOR OTHERS : 75 MM
CANT EXCESS (406(3))
75 MM

28. MAXIMUM VALUE OF Cex

29. TRANSITION CURVE
TRANSITION CURVE IS AN EASEMENT CURVE INTRODUCED BETWEEN STRAIGHT & CURVED TRACK TO FACILITATE GRADUAL CHANGE OF VERSINES & SUPERELEVATION
ON INDIAN RAILWAYS, IT IS CUBICAL PARABOLA WITH THE EQUATION:
Y = KX3
THE SPIRAL, WHICH CHANGES THE DIRECTION ANGLE UNIFORMLY, IS THE IDEAL TRANSITION
THERE IS NOT MUCH DIFFERENCE IN THE LAYOUT OF THE SPIRAL AND CUBIC PARABOLA UNTIL THE DEFLECTION FROM STRAIGHT IS APPROX 4 M.

30. VERSINE AND CANT DIAGRAM OF A CURVE
TRANSITION CURVES
CURVATURE VARIES UNIFORMLY WITH DISTANCE
VERSINES VARY UNIFORMLY
CANT VARIES UNIFORMLY V Ca
VERSINE AND CANT DIAGRAM OF A CURVE

31. SHIFT ON TRANSITION CURVE
CIRCULAR
CURVE WITHOUT TRANSITION
CIRCULAR
CURVE WITH TRANSITION
EXTENDED CIRCULAR CURVE
d2y/dx2=k x. dy/dx=kx2/2. y = kx3/6. at L, 1/R=kL. k=1/RL. Y=x3/6RL. At L/2, S/2= L2/48R. S=L2/24R C D B E F G A S
TRANSITION CURVE
TANGENT H
S/2
L/2

32. LENGTH OF TRANSITION COMFORT CRITERIA
RATE OF CHANGE OF ULA LESS THAN 0.03g
RATE OF CHANGE OF Cd <0.03G
FIXED AS 35 mm /sec
RATE OF CHANGE OF Ca IS JUST NOTICEABLE AT 65 TO 75mm/sec BUT FIXED AS 35 mm /sec
UNDER EXCEPTIONAL CIRCUMSTANCES IT CAN BE INCREASED TO 55 mm /sec

33. LENGTH OF TRANSITION SAFETY CRITERIA
CANT GRADIENT CAUSES TWIST IN TRACK
LIMITED TO 1.4 mm/m
CANT GRADIENT – 1 IN 720
IN EXCEPTIONAL CASES IT CAN BE 1 IN 360
FUTURE LAYOUTS WITH 1 IN 1200

34. LENGTH OF TRANSITION LENGTH OF TRANSITION WILL BE MAX. OF
L1 =0.008 Ca*Vm
L2 =0.008 Cd*Vm
L3 =0.72 Ca
MINIMUM LENGTH OF TRANSITION WILL BE MAX OF
2/3RD OF L1
2/3RD OF L2
½ OF L3

35. SUITABILITY OF CUBIC PARABOLA AS TRANSITION
CUBIC PARABOLA IS SIMPLE TO LAY AND MAINTAIN AS VERSINES VARY LINEARLY.
THERE IS SHOCK AT ENTRY AND EXIT OF THE CUBIC PARABOLA DUE TO SUDDEN CHANGE IN THE VERSINES.
END CORRECTION IS TO BE APPLIED AT THE 0 STATION AND SECOND LAST STATION.

36. SUITABILITY OF CUBIC PARABOLA AS TRANSITION
FOR SPEEDS ABOVE 130 KMPH, CUBIC PARABOLA IS NOT SUITABLE AS THE EFFECT OF END CONDITIONS GETS PRONOUNCED.
ALTERNATIVE SHAPES ARE
FOURTH ORDER PARABOLA
HALF SINE CURVE
FULL SINE CURVE

37. V
VIRTUAL TRANSITION

38. VIRTUAL TRANSITION
IF THERE IS NO SPACE FOR TRANSITION, THE BOGIE LENGTH BECOMES THE TRANSITION VIRTUALLY
For BG M
For MG-13.7 M

39. COMPOUND AND REVERSE CURVES
For Reverse Curve: Length of transition will be MAX. of
L =0.008*(Ca1+Ca2)*Vm
L =0.008 (Cd1+Cd2)*Vm
L =0.72 (Ca1+Ca2)

40. COMPOUND AND REVERSE CURVES
For Compound Curves:
Length of transition
will be MAX. of
L =0.008 (Ca1-Ca2)*Vm
L =0.008 (Cd1-Cd2)*Vm
L =0.72 (Ca1-Ca2)
If length is coming less than virtual transition then common transition is deleted and the cant is run out on the length of virtual transition

41. REVERSE CURVES
For high speeds in Group A and B routes a straight of 50m length shall be kept
Otherwise, increase the transition length to eliminate the straight
If neither of the above two are possible than speed restriction of 130 KMPH on BG

42. Vertical Curves

43. VERTICAL CURVES ROUTE MIN.RADIUS (m) A 4000 B 3000 C,D,E &MG 2500
VERTICAL CURVES ARE PROVIDED AT THOSE LOCATIONS WHERE ALGEBRAIC DIFFERENCE BETWEEN THE GRADES IS EQUAL TO OR MORE THAN 0.4% (4mm/m)
ROUTE
MIN.RADIUS (m) A
4000 B
3000
C,D,E &MG
2500

44. Extra Clearances on curves

45. CLEARANCE ON CURVES- LEANH Ca
L = H*Ca/G

46. ADDITIONAL ALLOWANCE DUE TO SWAY
INSIDE
¼ OF LEAN DUE TO SUPERELEVATION
OUTSIDE
NIL

47. Extra Clearances on Curves

48. ADDITIONAL CLEARANCES ON CURVES
OVER THROW Vo
CENTRE LINE
OF COACH
END THROW VE
BOGIE CENTRE
14.785
21.340
CENTRE LINE
OF TRACK

49. EXTRA CLEARANCE DUE TO CURVATURE
PLATFORMS/ STRUCTURES
INSIDE OF CURVE
(VO+L+S-51)
OUTSIDE OF CURVE
(VE-25)mm
BETWEEN ADJACENT TRACKS
VO+ VE+2*(L/4)
In new works, if c/c track is 5300 mm, extra clearance is to be provided for curves beyond 5 degrees only.

50. TURNOUTS TAKING OFF FROM CURVES
SIMILAR FLEXURE
EQUIVALENT RADIUS
Re=Rm*Rs/(Rm+Rs)
Rm: MAIN LINE RADIUS
Rs: SWITCH RADIUS
Re: EQUIVALENT RADIUS

51. CONTRARY FLEXURE
EQUIVALENT RADIUS
Re=Rm*Rs/(Rm-Rs)

52. TURNOUTS OF CONTRARY FLEXURE
CURVES OF CONTRARY FLEXURE
EQUILIBRIUM SUPER ELEVATION FOR TURNOUT SIDE
C= G * V2 / 127 * R; V: SPEED KPMH FOR TURNOUT, R: Metres, C: MM, G: MM
NEGATIVE SUPERELEVATION FOR TURNOUT IS 75-C

53. CHECK RAILS ON CURVES PARA 426 IRPWM CL. 4, CHAPTER 1, BG SOD
TOO MUCH WEAR ON OUTER RAIL IN SHARP CURVES
CHECK RAIL
CLEAR-ANCE
PARA 426 IRPWM
CL. 4, CHAPTER 1, BG SOD
ON CURVES OF 8 DEGREES & ABOVE
MIN. CLEARANCE = 44 mm
CLEARANCE = 44 + (G-1676)/2

54. IRPWM PROVISIONS INSPECTIONS
AEN TO INSPECT ONE CURVE IN EACH PWI JURISDICTION EVERY QUARTER (107(4))
PWI INCHARGE AND HIS ASSISTANT SHALL INSPECT CURVES ONCE IN SIX MONTHS ALTERNATELY EXCEPT AT GROUP A AND B ROUTES WHERE INSPECTION IS TO BE DONE ONCE IN FOUR MONTHS (124(4), 139(4))
FOR CONCRETE SLEEPER TRACK, ONCE IN SIX MONTHS BY ROTATION (124A, 139A)

55. IRPWM PROVISIONS LATERAL WEAR ON RAILS IN CURVES (301(b)(iv))
GROUP A AND B : 8 MM
GROUP C AND D : 10 MM
MINIMISING WEAR ON OUTER RAIL (427(2))
RAIL FLANGE LUBRICATORS SHALL BE PROVIDED IN CURVES LESS THAN 600 M RADIUS. FIRST ONE SHALL BE AHEAD OF THE CURVE

56. IRPWM PROVISIONS CUT IN RAILS IN SWR TRACK (424 )
TO MAKE THE JOINTS SQUARE WHEN THE GAIN OF INNER RAIL BECOMES EQUAL TO PITCH OF THE FIRST BOLT HOLE
GAIN OF INNER RAIL GIVEN BY: d= LG/R, L IS LENGTH OF CURVE.
ON CURVES LESS THAN 400 M RADIUS, THE JOINTS SHALL BE LAID STAGGERED

57. IRPWM PROVISIONS AFTER RELAYING
VERSINE VARIATION OVER THEORETICAL (316)
600 M RADIUS AND MORE: 5 MM
LESS THAN 600 M RADIUS: 10 MM
GAUGE IN TRACK ON CURVES (403(1))
MORE THAN 350 M RADIUS: -5 MM TO + 3 MM
LESS THAN 350 M RADIUS: +10 MM

58. IRPWM PROVISIONS IN SERVICE PARAMETERS GAUGE (224(2)(e)(v))
ON STRAIGHT : (-) 6 MM TO (+) 6 MM
ON CURVES RADIUS 350 M OR MORE: (-) 6 MM TO (+) 15 MM
ON CURVES RADIUS LESS THAN 350 M: UPTO (+) 20 MM

59. IRPWM PROVISIONS
IN SERVICE TOLERANCES (SPEED >100 KMPH, <140 KMPH)
TWIST (607(2)(iii))
ON STRAIGHT/CURVES
NORMAL 2 mm/M
ISOLATED 3.5 mm/M
ON TRANSITIONS
NORMAL 1mm/M
ISOLATED 2.1 mm/M

60. IRPWM PROVISIONS ALIGNMENT (607(2)(i)) 7.5 M CHORD
IN SERVICE TOLERANCES (SPEED >100 KMPH, <140 KMPH)
ALIGNMENT (607(2)(i)) 7.5 M CHORD
ON STRAIGHT
NORMAL 5 mm
ISOLATED 10 mm
ON CURVES
NORMAL ±5 mm OVER AVERAGE VERSINE
ISOLATED ±7 mm OVER AVERAGE VERSINE
CHORD TO CHORD > 10 mm

61. IRPWM PROVISIONS RADIUS OF CURVE (401(1))
R = 125C2/V
FOR MEASURING VERSINES (401(3))
NORMAL: 20 M CHORD OVERLAPPING WITH STATIONS AT 10 M
P & Cs: 6 M CHORD OVERLAPPING WITH STATIONS AT 1.5 M
INNER RAIL IS TAKEN AS REFERENCE AND OUTER RAIL IS RAISED BY AMOUNT OF SUPERELEVATION(402)

62. IRPWM PROVISIONS SAFE SPEED ON A CURVE (405(1)(a))
V = 0.27 √R (Ca + Cd)
NON TRANSITIONED CURVE (405(2)(a))
LENGTH OF TRANSITION : 14.6 M
ON OUTSIDE OF CURVES, EXTRA WIDTH OF BALLAST TO BE PROVIDED
150 MM: FOR LWR TRACK (ANN 2/11 AND 2/12)
95 MM: ON FISH PLATED TRACK, RADIUS LESS THAN 400 M (ANN 2/13 )

63. LEAD CURVE FOLLOWING TURNOUT
MINIMUM RADIUS OF LEAD AND TURN IN CURVE (410(2)) : 350 M
MIN RADIUS IN LEAD 220 M, IN EXCEPTIONAL CIRCUMSTANCES (WITH PSC/ST SLEEPERS AND FULL ML BALLAST PROFILE)
THERE SHALL BE NO CHANGE IN SUPERELEVATION 20 M ON EITHER SIDE OF THE TOE OF SWITCH AND NOSE OF CROSSING
TURNOUT FOLLOWED BY REVERSE CURVE, CHANGE IN CANT BEHIND CROSSING IS PERMITTED. CANT LIMITED TO 65 MM, RUN OUT AT 2.8 MM/M

64. LEAD CURVE FOLLOWING TURNOUT
The variation in versines on two successive stations in lead curve and turn in curve portions should not be more than 4mm. And versine at each station should also not be beyond ± 3mm. from its designed value (237(4)(d))

65. LIMITING RADII BG : 175 M MG: 109 M NG: 44 M
Item 2, Chapter I, Schedule I of SOD

66. IRPWM PROVISIONS CURVE BOARDS(409(1)) CURVE POSTS (409(2))
CURVE BOARD SHALL BE PROVIDED AT TANGENT POINT ON OUTSIDE OF THE CURVE
CURVE POSTS (409(2))
RAIL POSTS INDICATING BEGINNING AND END OF TRANSITION, PAINTED RED AND WHITE SHALL BE PROVIDED

67. IRPWM PROVISIONS SUPERELEVATION MARKING ON RAILS (409(3))
VALUE OF SUPERELEVATION SHALL BE MARKED ON THE WEB OF INSIDE RAIL AT VERSINE STATIONS IN TRANSITION PORTION
VALUE AS ABOVE SHALL BE MARKED AT THE BEGINNING AND END OF THE CIRCULAR CURVE
FOR LONGER CURVES, THE VALUE OF SUPERELEVATION SHALL BE REPEATED AS ABOVE AT INTERVALS NOT EXCEEDING 250 M

68. THANK YOU